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Volume
The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically. One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space. Volume is commonly presented in units such as mL or cm3 (milliliters or cubic centimeters). Volumes of some simple shapes, such as regular, straight-edged and circular shapes can be easily calculated using arithmetic formulas. More complicated shapes can be calculated by integral calculus if a formula exists for its boundary. The volume of any shape can be determined by displacement. In differential geometry, volume is expressed by means of the volume form, and is an important global Riemannian invariant. Volume is a fundamental parameter in thermodynamics and it is conjugate to pressure. Traditional cooking measures In the UK, a tablespoon can also be five fluidrams (about 17.76 mL). Volume formulas The units of volume depend on the units of length. If the lengths are in meters, the volume will be in cubic meters. : For their volume formulas, see the articles on tetrahedron and parallelepiped. Volume formula derivation Sphere The volume of a sphere is the integral of infinitesimal circular slabs of thickness dx . The calculation for the volume of a sphere with center 0 and radius r is as follows. The radius of the circular slabs is y=\sqrt{r^2-x^2} The surface area of the circular slab is \pi y^2 . The volume of the sphere can be calculated as \int\limits_{-r}^r\pi(r^2-x^2)\,dx=\int\limits_{-r}^r\pi r^2\,dx-\int\limits_{-r}^r\pi x^2\,dx :Now \int\limits_{-r}^r\pi r^2\,dx=2\pi r^3 :and \int\limits_{-r}^r\pi x^2\,dx=\frac{2\pi}{3}r^3 Combining yields \left(2-\frac{2}{3}\right)\pi r^3=\frac{4\pi}{3}r^3 This formula can be derived more quickly using the formula for the sphere's surface area, which is 4\pi r^2 . The volume of the sphere consists of layers of infinitesimal spherical slabs, and the sphere volume is equal to \int\limits_0^r 4\pi r^2\,dr=\frac{4\pi}{3}r^3 Cone The volume of a cone is the integral of infinitesimal circular slabs of thickness dx . The calculation for the volume of a cone of height h , whose base is centered at (0,0) with radius r is as follows. The radius of each circular slab is \begin{cases}r&x=0\\0&x=h\end{cases} , and varying linearly in between—that is, r\left(1-\frac{x}{h}\right) The surface area of the circular slab is then \pi\left(r(1-\frac{x}{h})\right)^2=\pi r^2\frac{(h-x)^2}{h^2} The volume of the cone can then be calculated as \int\limits_0^h\pi r^2\frac{(h-x)^2}{h^2}dx And after extraction of the constants: \frac{\pi r^2}{h^2}\int\limits_0^h (h-x)^2dx Integrating gives us \frac{\pi r^2}{h^2}\left(\frac{h^3}{3}\right)=\frac{\pi}{3}r^2h See also * Area * Conversion of units * Density * Dimensional weight * Dimensioning * Length * Mass * Orders of magnitude (volume) * Volume form * Weight External links * Fortran code for finding volumes of various shapes * Unit converter tool also for volume af:Volume ar:حجم be-x-old:Аб'ём bg:Обем ca:Volum cs:Objem da:Rumfang de:Volumen el:Όγκος es:Unidades de volumen eo:Volumeno eu:Bolumen (espazioa) fa:حجم fr:Volume ko:부피 hi:आयतन hr:Obujam id:Volume it:Volume he:נפח lb:Volumen lt:Tūris lv:Tilpums hu:Térfogat mk:Волумен ms:Isi padu nl:Inhoud ja:体積 no:Volum nn:Volum uz:Hajm pl:Objętość pt:Volume ro:Volum qu:P'ulin ru:Объём simple:Volume sk:Objem (matematika) sl:Prostornina szl:Uobjyntość sr:Запремина fi:Tilavuus sv:Volym ta:கனவளவு th:ปริมาตร vi:Thể tích tr:Hacim uk:Об'єм ur:حجم zh:体积 Category:Geometry Category:Formulas